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Properties

Label 2-336-112.109-c1-0-3
Degree $2$
Conductor $336$
Sign $0.117 - 0.993i$
Analytic cond. $2.68297$
Root an. cond. $1.63797$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.30 − 0.539i)2-s + (0.965 − 0.258i)3-s + (1.41 + 1.41i)4-s + (−3.17 − 0.850i)5-s + (−1.40 − 0.182i)6-s + (−2.50 + 0.865i)7-s + (−1.09 − 2.60i)8-s + (0.866 − 0.499i)9-s + (3.68 + 2.82i)10-s + (0.976 + 3.64i)11-s + (1.73 + 0.995i)12-s + (3.52 + 3.52i)13-s + (3.73 + 0.217i)14-s − 3.28·15-s + (0.0220 + 3.99i)16-s + (−2.06 + 3.57i)17-s + ⋯
L(s)  = 1  + (−0.924 − 0.381i)2-s + (0.557 − 0.149i)3-s + (0.709 + 0.705i)4-s + (−1.41 − 0.380i)5-s + (−0.572 − 0.0745i)6-s + (−0.944 + 0.327i)7-s + (−0.386 − 0.922i)8-s + (0.288 − 0.166i)9-s + (1.16 + 0.892i)10-s + (0.294 + 1.09i)11-s + (0.500 + 0.287i)12-s + (0.978 + 0.978i)13-s + (0.998 + 0.0580i)14-s − 0.848·15-s + (0.00550 + 0.999i)16-s + (−0.500 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.117 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.117 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.117 - 0.993i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.117 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.373324 + 0.331910i\)
\(L(\frac12)\) \(\approx\) \(0.373324 + 0.331910i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.30 + 0.539i)T \)
3 \( 1 + (-0.965 + 0.258i)T \)
7 \( 1 + (2.50 - 0.865i)T \)
good5 \( 1 + (3.17 + 0.850i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.976 - 3.64i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-3.52 - 3.52i)T + 13iT^{2} \)
17 \( 1 + (2.06 - 3.57i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.567 - 2.11i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.113 + 0.0653i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.42 + 3.42i)T + 29iT^{2} \)
31 \( 1 + (3.48 - 6.03i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (9.27 + 2.48i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 11.3iT - 41T^{2} \)
43 \( 1 + (-3.06 + 3.06i)T - 43iT^{2} \)
47 \( 1 + (-4.39 - 7.60i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.00 + 11.2i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.701 + 2.61i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.852 + 3.18i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-7.58 + 2.03i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 8.65iT - 71T^{2} \)
73 \( 1 + (13.9 + 8.02i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.80 - 6.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.578 + 0.578i)T + 83iT^{2} \)
89 \( 1 + (-1.27 + 0.735i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 3.94T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.78499821639274361026386946027, −10.84521661621916741769291827663, −9.689431131796416845593764977521, −8.900441567099203648310282910017, −8.248671971242731487810713669703, −7.22466751203291548788431005691, −6.44966446492438442580909715496, −4.15457452192950342146826898485, −3.49089665057268427014343001146, −1.79126293840655723566691674823, 0.43941760990752149084362134348, 2.99158062753395675935902534197, 3.81597933955625316938691392353, 5.71459596339748596605586085912, 6.93532026983349941311157684089, 7.56876637750198627187004361437, 8.610810668429408505923589991103, 9.144125956077059205345555647574, 10.53472034517428902047373172075, 10.98900260999012858328759990256

Graph of the $Z$-function along the critical line